Incompressible Materials
Incompressible Materials
•Many nonlinear problems involve
incompressible materials
( = 0.5) and nearly incompressible
materials ( > 0.475).
•Rubber
•Metals at large plastic
strains
–Conventional finite element
meshes often exhibit overly
stiff behavior due to volumetric
locking, which is most severe
when these materials are
highly confined.
overly stiff behavior of an elasticplastic material with volumetric locking
correct behavior of an elasticplastic material
Example of the effect of volumetric
locking
Incompressible Materials
–The cause of volumetric locking is that each integration point’s volume must
remain almost constant, overconstraining the kinematically admissible
displacement field.
•For example, in a refined three-dimensional mesh of 8-node hexahedra,
there is—on average—1 node with 3 degrees of freedom per element.
每个单元平均只有1个有三个自由度的节点
•The volume at each integration point must remain fixed.
•Fully integrated hexahedra use 8 integration points per element; thus,
in this example we have as many as 8 constraints per element, but only
3 degrees of freedom are available to satisfy these constraints. 每个单元
有8个约束，以至于产生体积锁死。
•The mesh is overconstrained—it “locks.”
–Volumetric locking is most pronounced in fully integrated elements.
–Reduced-integration elements have fewer volumetric constraints.
•Reduced integration effectively eliminates volumetric locking in many
problems with nearly incompressible material.
Incompressible Materials
–Fully incompressible materials modeled with solid elements
must use the “hybrid” formulation (elements whose names
end with the letter “H”).
•In this formulation the pressure stress is treated as an
independently interpolated basic solution variable,
coupled to the displacement solution through the
constitutive theory.
•Hybrid elements introduce more variables into the
problem to alleviate the volumetric locking problem. The
extra variables also make them more expensive.
•The ABAQUS element library includes hybrid versions
of all continuum elements (except plane stress elements,
where they are not needed).
Incompressible Materials
–Hybrid elements are only necessary for:
•以不可压缩材料为主的网格，如橡胶材料。All
meshes with strictly incompressible materials, such as
rubber.
•精密的网格，使用减缩积分仍然有locking的网格，比
如弹塑性材料完全进入塑性阶段
Refined meshes of reduced-integration elements that
still show volumetric locking problems. Such problems
are possible with elastic-plastic materials strained far
into the plastic range.
–即使使用了hybrid单元一次三角形或者四面体单元仍然有
过度约束。因此建议这类单元使用的比例要小，可以作为六
面体单元的“填充物”使用。Even with hybrid elements a
mesh of first-order triangles and tetrahedra is
overconstrained when modeling fully incompressible
materials. Hence, these elements are recommended only
for use as “fillers” in quadrilateral or brick-type meshes with
such material.
Mesh Generation
Mesh Generation
•Quad/Hex vs. Tri/Tet Elements
–Of particular importance when
generating a mesh is the decision
regarding whether to use quad/hex
or tri/tet elements.
–Quad/hex elements should be
used wherever possible.
•They give the best results for
the minimum cost.
•When modeling complex
geometries, however, the
analyst often has little choice
but to mesh with triangular
and tetrahedral elements.
Turbine blade with platform modeled
with tetrahedral elements
Mesh Generation
–First-order tri/tet elements (CPE3, CPS3, CAX3, C3D4,
C3D6) are poor elements; they have the following problems:
•Poor convergence rate.
–They typically require very fine meshes to produce
good results.
•Volumetric locking with incompressible or nearly
incompressible materials, even using the “hybrid”
formulation.
–These elements should be used only as fillers in regions far
from any areas where accurate results are needed.
Mesh Generation
– “Regular” second-order tri/tet
elements (CPE6, CPS6, CAX6,
C3D10) cannot be used to
model contact.
• Under uniform pressure
the contact forces are
significantly different at
the corner and midside
nodes.
– For small-displacement
problems without contact
these elements provide
reasonable results.
Equivalent nodal forces
created by uniform
pressure on the face of a
regular second-order
tetrahedral element
Mesh Generation
–Modified second-order tri/tet elements (C3D10M, etc.) alleviate
the problems of other tri/tet elements.
•Good convergence rate—close to convergence rate of
second-order quad/hex elements.
•Minimal shear or volumetric locking.
–Can be used to model incompressible or nearly
incompressible materials in the hybrid formulation
(C3D10MH).
•These elements are robust during finite deformation.
•Uniform contact pressure allows these elements to model
contact accurately.
•Use them!
Mesh Generation
•Mesh refinement and convergence
–Use a sufficiently refined mesh to ensure that the results from
your ABAQUS simulation are adequate.
•Coarse meshes tend to yield inaccurate results.
•The computer resources required to run your job increase
with the level of mesh refinement.
–It is rarely necessary to use a uniformly refined mesh throughout
the structure being analyzed.
•Use a fine mesh only in areas of high gradients and a
coarser mesh in areas of low gradients.
–You can often predict regions of high gradients before
generating the mesh.
•Use hand calculations, experience, etc.
•Alternatively, you can use coarse mesh results to identify
high gradient regions.
Mesh Generation
–Some recommendations:
•Minimize mesh distortion as much as possible.
•A minimum of four quadratic elements per 90o
should be used around a circular hole.
•A minimum of four elements should be used through
the thickness of a structure if first-order, reducedintegration solid elements are used to model bending.
•Other guidelines can be developed based on
experience with a given class of problem.
Mesh Generation
–It is good practice to perform a mesh convergence study.
•Simulate the problem using progressively finer meshes,
and compare the results.
–The mesh density can be changed very easily
using ABAQUS/CAE since the definition of the
analysis model is based on the geometry of the
structure.
–This will be discussed further in the next lecture.
•When two meshes yield nearly identical results, the
results are said to have “converged.”
–This provides increased confidence in your results.
Solid Element
Selection Summary
Solid Element Selection Summary
Class of problem
Best element choice
Avoid using
General contact
between
deformable bodies
First-order quad/hex
Second-order quad/hex
Contact with bending
Incompatible mode
First-order fully integrated
quad/hex or secondorder quad/hex
Bending (no contact)
Second-order
quad/hex
First-order fully integrated
quad/hex
Stress concentration
Second-order
First-order
Nearly incompressible
(n>0.475 or large
strain plasticity
epl>10%)
First-order elements
or second-order
reducedintegration
elements
Second-order fully
integrated
Solid Element Selection Summary
Class of problem
Completely
incompressible (rubber
n = 0.5)
Bulk metal forming (high
mesh distortion)
Best element choice
Avoid using
Hybrid quad/hex, first-order if large
deformations are anticipated
First-order reduced-integration
quad/hex
Complicated model
geometry (linear
material, no contact)
Second-order quad/hex if possible (if
not overly distorted) or secondorder tet/tri (because of meshing
difficulties)
Complicated model
geometry (nonlinear
problem or contact)
First-order quad/hex if possible (if not
overly distorted) or modified
second-order tet/tri (because of
meshing difficulties)
Natural frequency (linear
dynamics)
Second-order
Nonlinear dynamic
(impact)
First-order
Second-order
quad/hex
Second-order